Let \(p\)  be a prime such that \(\frac{a(tp^{2\nu })}{\varepsilon (p)^\nu }\ge 0\)  for all but finitely many \(\nu \ge 0\) . Thus, by Landau's theorem [1]}, the series in the left-hand side of (REF ) either (a) converges for all \(s\in \mathbb {C}\)  or (b) has a singularity at the real point of its line of convergence. It is clear that the alternative (a) cannot occur, since the right-hand side of (REF ) has a pole for \(p^{s}=\alpha ^{\prime }_p\)  or \(p^{s}=\beta ^{\prime }_p.\) 
Thus the alternative (b) must hold, therefore \(\alpha ^{\prime }_p\)  or \(\beta ^{\prime }_p\)  must be real. On the other hand since
\(\frac{\lambda _p}{\varepsilon (p)}\in \mathbb {R}\) , it follows \(\alpha _p^{\prime }=\overline{\beta _p^{\prime }}\) . Moreover, by Deligne's theorem [2]}
we have \(|\alpha ^{\prime }_p|=|\beta ^{\prime }_p|=p^{k-1/2}\) . Consequently
\(\lambda _p=\pm 2 p^{k-1/2}\varepsilon (p),\) 
